No.
ONLY use the CPU fan connector for your CPU fan.As for the rest of what you said: MOST (ok, only every one that I've ever purchased) add-on fans that are thermally speed controlled HAVE THE THERMAL CONTROLLER BUILT INTO THEM, and are three-pin.
If you bought a four-pin fan, then you have to have a board with four-pin fan connectors to make use of
pulse-width modulation and connecting them via special on-board connectors so the board can manage the fan through special circuitry.
I doubt very much that you bought a
thermostatic fan, so I won't go into that here....
Take it back and get a three-pin fan with thermal control, or with an external speed controlling switch.